The 14-dimensional exceptional Lie group \(G_2\)  is the subgroup of \(\mathrm {SO}(7)\)  preserving the alternating 3-form \(\epsilon \)  on \(\mathbb {R}^7\)  defined by identifying \(\mathbb {R}^7\)  with the pure-imaginary octonions and setting \(\epsilon (a,b,c) = (ab)c - a(bc) \in \mathbb {R}\)  [1]}. Since \(\mathrm {SO}(7)\)  acts by linear symplectic automorphisms of \(\mathbb {R}^{0|7}\) , we get an induced symplectic action of \(G_2\) .
